Theorem supminfxrrnmpt 39701

Description: The indexed supremum of a set of reals is the negation of the indexed infimum of that set's image under negation. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Hypotheses

Ref	Expression
supminfxrrnmpt.x	⊢ Ⅎ𝑥𝜑
supminfxrrnmpt.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)

Assertion

Ref	Expression
supminfxrrnmpt	⊢ (𝜑 → sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) = -𝑒inf(ran (𝑥 ∈ 𝐴 ↦ -𝑒𝐵), ℝ*, < ))

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem supminfxrrnmpt

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		supminfxrrnmpt.x	. . . 4 ⊢ Ⅎ𝑥𝜑
2		eqid 2622	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
3		supminfxrrnmpt.b	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)
4	1, 2, 3	rnmptssd 39385	. . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ*)
5	4	supminfxr2 39699	. 2 ⊢ (𝜑 → sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) = -𝑒inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}, ℝ*, < ))
6		xnegex 12039	. . . . . . . . . . . 12 ⊢ -𝑒𝑦 ∈ V
7	2	elrnmpt 5372	. . . . . . . . . . . 12 ⊢ (-𝑒𝑦 ∈ V → (-𝑒𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 -𝑒𝑦 = 𝐵))
8	6, 7	ax-mp 5	. . . . . . . . . . 11 ⊢ (-𝑒𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 -𝑒𝑦 = 𝐵)
9	8	biimpi 206	. . . . . . . . . 10 ⊢ (-𝑒𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) → ∃𝑥 ∈ 𝐴 -𝑒𝑦 = 𝐵)
10		eqid 2622	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 ↦ -𝑒𝐵) = (𝑥 ∈ 𝐴 ↦ -𝑒𝐵)
11		xnegneg 12045	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℝ* → -𝑒-𝑒𝑦 = 𝑦)
12	11	eqcomd 2628	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℝ* → 𝑦 = -𝑒-𝑒𝑦)
13	12	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℝ* ∧ -𝑒𝑦 = 𝐵) → 𝑦 = -𝑒-𝑒𝑦)
14		xnegeq 12038	. . . . . . . . . . . . . . . 16 ⊢ (-𝑒𝑦 = 𝐵 → -𝑒-𝑒𝑦 = -𝑒𝐵)
15	14	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℝ* ∧ -𝑒𝑦 = 𝐵) → -𝑒-𝑒𝑦 = -𝑒𝐵)
16	13, 15	eqtrd 2656	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℝ* ∧ -𝑒𝑦 = 𝐵) → 𝑦 = -𝑒𝐵)
17	16	ex 450	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ* → (-𝑒𝑦 = 𝐵 → 𝑦 = -𝑒𝐵))
18	17	reximdv 3016	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ* → (∃𝑥 ∈ 𝐴 -𝑒𝑦 = 𝐵 → ∃𝑥 ∈ 𝐴 𝑦 = -𝑒𝐵))
19	18	imp 445	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℝ* ∧ ∃𝑥 ∈ 𝐴 -𝑒𝑦 = 𝐵) → ∃𝑥 ∈ 𝐴 𝑦 = -𝑒𝐵)
20		simpl 473	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℝ* ∧ ∃𝑥 ∈ 𝐴 -𝑒𝑦 = 𝐵) → 𝑦 ∈ ℝ*)
21	10, 19, 20	elrnmptd 39366	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℝ* ∧ ∃𝑥 ∈ 𝐴 -𝑒𝑦 = 𝐵) → 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝑒𝐵))
22	9, 21	sylan2 491	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℝ* ∧ -𝑒𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝑒𝐵))
23	22	ex 450	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ* → (-𝑒𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) → 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝑒𝐵)))
24	23	rgen 2922	. . . . . . 7 ⊢ ∀𝑦 ∈ ℝ* (-𝑒𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) → 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝑒𝐵))
25		rabss 3679	. . . . . . . 8 ⊢ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} ⊆ ran (𝑥 ∈ 𝐴 ↦ -𝑒𝐵) ↔ ∀𝑦 ∈ ℝ* (-𝑒𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) → 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝑒𝐵)))
26	25	biimpri 218	. . . . . . 7 ⊢ (∀𝑦 ∈ ℝ* (-𝑒𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) → 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝑒𝐵)) → {𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} ⊆ ran (𝑥 ∈ 𝐴 ↦ -𝑒𝐵))
27	24, 26	ax-mp 5	. . . . . 6 ⊢ {𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} ⊆ ran (𝑥 ∈ 𝐴 ↦ -𝑒𝐵)
28	27	a1i 11	. . . . 5 ⊢ (𝜑 → {𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} ⊆ ran (𝑥 ∈ 𝐴 ↦ -𝑒𝐵))
29		nfcv 2764	. . . . . . . 8 ⊢ Ⅎ𝑥-𝑒𝑦
30		nfmpt1 4747	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
31	30	nfrn 5368	. . . . . . . 8 ⊢ Ⅎ𝑥ran (𝑥 ∈ 𝐴 ↦ 𝐵)
32	29, 31	nfel 2777	. . . . . . 7 ⊢ Ⅎ𝑥-𝑒𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)
33		nfcv 2764	. . . . . . 7 ⊢ Ⅎ𝑥ℝ*
34	32, 33	nfrab 3123	. . . . . 6 ⊢ Ⅎ𝑥{𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}
35		xnegeq 12038	. . . . . . . 8 ⊢ (𝑦 = -𝑒𝐵 → -𝑒𝑦 = -𝑒-𝑒𝐵)
36	35	eleq1d 2686	. . . . . . 7 ⊢ (𝑦 = -𝑒𝐵 → (-𝑒𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ -𝑒-𝑒𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)))
37	3	xnegcld 12130	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝑒𝐵 ∈ ℝ*)
38		xnegneg 12045	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ* → -𝑒-𝑒𝐵 = 𝐵)
39	3, 38	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝑒-𝑒𝐵 = 𝐵)
40		simpr 477	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
41	2, 40, 3	elrnmpt1d 39435	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
42	39, 41	eqeltrd 2701	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝑒-𝑒𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
43	36, 37, 42	elrabd 3365	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝑒𝐵 ∈ {𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)})
44	1, 34, 10, 43	rnmptssdf 39469	. . . . 5 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ -𝑒𝐵) ⊆ {𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)})
45	28, 44	eqssd 3620	. . . 4 ⊢ (𝜑 → {𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} = ran (𝑥 ∈ 𝐴 ↦ -𝑒𝐵))
46	45	infeq1d 8383	. . 3 ⊢ (𝜑 → inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}, ℝ*, < ) = inf(ran (𝑥 ∈ 𝐴 ↦ -𝑒𝐵), ℝ*, < ))
47	46	xnegeqd 39664	. 2 ⊢ (𝜑 → -𝑒inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}, ℝ*, < ) = -𝑒inf(ran (𝑥 ∈ 𝐴 ↦ -𝑒𝐵), ℝ*, < ))
48	5, 47	eqtrd 2656	1 ⊢ (𝜑 → sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) = -𝑒inf(ran (𝑥 ∈ 𝐴 ↦ -𝑒𝐵), ℝ*, < ))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  Ⅎwnf 1708   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  {crab 2916  Vcvv 3200   ⊆ wss 3574   ↦ cmpt 4729  ran crn 5115  supcsup 8346  infcinf 8347  ℝ^*cxr 10073   < clt 10074  -_𝑒cxne 11943

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-xneg 11946

This theorem is referenced by:  liminfvalxr  40015